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Area Properties of Strictly Convex Curves mathematics Article Area Properties of Strictly Convex Curves Dong-Soo Kim 1, Young Ho Kim 2,* and Yoon-Tae Jung 3 1 Department of Mathematics, Chonnam National University, Gwangju 61186, Korea; [email protected] 2 Department of Mathematics, Kyungpook National University, Daegu 41566, Korea 3 Department of Mathematics, Chosun University, Gwangju 61452, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-53-950-5882 Received: 28 March 2019; Accepted: 21 April 2019; Published: 29 April 2019 Abstract: We study functions defined in the plane E2 in which level curves are strictly convex, and investigate area properties of regions cut off by chords on the level curves. In this paper we give a partial answer to the question: Which function has level curves whose tangent lines cut off from a level curve segment of constant area? In the results, we give some characterization theorems regarding conic sections. Keywords: Archimedes; level curve; chord; conic section; strictly convex plane curve; curvature; equiaffine transformation 1. Introduction The most well-known plane curves are straight lines and circles, which are characterized as the plane curves with constant Frenet curvature. The next most familiar plane curves might be the conic sections: ellipses, hyperbolas and parabolas. They are characterized as plane curves with constant affine curvature ([1], p. 4). The conic sections have an interesting area property. For example, consider the following two −1 −1 ellipses given by Xk = g (k) and Xl = g (l) with l > k > 0, where x2 y2 g(x, y) = + , a, b > 0. a2 b2 For a fixed point p on Xk, we denote by A and B the points where the tangent to Xk at p meets Xl. Then the region D bounded by the ellipse Xl and the chord AB outside Xk has constant area independent of the point p 2 Xk. In order to give a proof, consider a transformation T of the plane E2 defined by p ! b/ ab 0 T = p . 0 a/ ab p p Then Xk and Xl are transformed to concentric circles of radius abk and abl, respectively; the tangent at p to the tangent at the corresponding point p0. Since the transformation T is equiaffine (that is, area preserving), a well-known property of concentric circles completes the proof. For parabolas and hyperbolas given by g(x, y) = y2 − 4ax, a 6= 0 and g(x, y) = x2/a2 − y2/b2, a, b > 0, respectively, it is straightforward to show that they also satisfy the above mentioned area properties. For a proof using 1-parameter group of equiaffine transformations, see [1], pp. 6–7. Conversely, it is reasonable to ask the following question. Question. Are there any other level curves of a function g : R2 ! R satisfying the above mentioned area property? Mathematics 2019, 7, 391; doi:10.3390/math7050391 www.mdpi.com/journal/mathematics Mathematics 2019, 7, 391 2 of 14 A plane curve X in the plane E2 is called ‘convex’ if it bounds a convex domain in the plane E2 [2]. A convex curve in the plane E2 is called ‘strictly convex’ if the curve has positive Frenet curvature k with respect to the unit normal N pointing to the convex side. We also say that a convex function f : R ! R is ‘strictly convex’ if the graph of f is strictly convex. 2 Consider a smooth function g : R ! R. We let Rg denote the set of all regular values of the function g. We suppose that there exists an interval Sg ⊂ Rg such that for every k 2 Sg, the level curve −1 2 Xk = g (k) is a smooth strictly convex curve in the plane E . We let Sg denote the maximal interval in Rg with the above property. If k 2 Sg, then there exists a maximal interval Ik ⊂ Sg such that each Xk+h with k + h 2 Ik lies in the convex side of Xk. The maximal interval Ik is of the form (k, a) or (b, k) according to whether the gradient vector rg points to the convex side of Xk or not. 2 2 2 2 As examples, consider the two functions gi : R ! R, i = 1, 2 defined by gi(x, y) = y + eia x i with positive constant a, ei = (−1) . Then, for the function g1 we have Rg1 = R − f0g, Sg1 = (0, ¥) or (−¥, 0), Ik = (k, ¥) if k > 0, and Ik = (−¥, k) if k < 0. For g2, we get Rg2 = Sg2 = (0, ¥) and Ik = (0, k) with k 2 Sg2 . For a fixed point p 2 Xk with k 2 Sg and a small h with k + h 2 Ik, we consider the tangent line t to Xk at p 2 Xk and the closest tangent line ` to Xk+h at a point v 2 Xk+h, which is parallel to the ∗ tangent line t. We let Ap(k, h) denote the area of the region bounded by Xk and the line ` (See Figure1). p p ∗ 2 2 Figure 1. Ap(1, −3/4) for p = (1, 3/2), v = (1/2, 3/4) and g(x, y) = x /4 + y . In [3], the following characterization theorem for parabolas was established. Proposition 1. We consider a strictly convex function f : R ! R and the function g : R2 ! R given by g(x, y) = y − f (x). Then, the following conditions are equivalent. ∗ 1. For a fixed k 2 R, Ap(k, h) is a function fk(h) of only h. 2. Up to translations, the function f (x) is a quadratic polynomial given by f (x) = ax2 with a > 0, and hence every level curve Xk of g is a parabola. In the above proposition, we have Rg = Sg = R and Ik = (k, ¥). In particular, Archimedes proved that every level curvep Xk (parabola) of the function g(x, y) = 2 2 ∗ y − ax in the Euclidean plane E satisfies Ap(k, h) = ch h for some constant c which depends only on the parabola [4]. In this paper, we investigate the family of strictly convex level curves Xk, k 2 Sg of a function g : R2 ! R which satisfies the following condition. ∗ ∗ (A ): For k 2 Sg with k + h 2 Ik, Ap(k, h) with p 2 Xk is a function fk(h) of only k and h. 2 In order to investigate the family of strictly convex level curves Xk, k 2 Sg of a function g : R ! R satisfying condition (A∗), first of all, in Section2 we introduce a useful lemma which reveals a relation between the curvature of level curves and the gradient of the function g (Lemma3 in Section2). Next, using Lemma3, in Section3 we establish the following characterizations for conic sections. Mathematics 2019, 7, 391 3 of 14 Theorem 1. Let f : R ! R be a smooth function. We let g denote the function defined by g(x, y) = ya − f (x), 2 where a is a nonzero real number with a 6= 1. Suppose that the level curves Xk(k 2 Sg) of g in the plane E are strictly convex. Then the following conditions are equivalent. 1. The function g satisfies (A∗). 3 2. For k 2 Sg, k(p)jrg(p)j = c(k) is constant on Xk, where k(p) denotes the curvature of Xk at p 2 Xk. 3. We have a = 2 and the function f is a quadratic function. Hence, each Xk is a conic section. In case the function f (− f , resp.) is itself a non-negative strictly convex function, Theorem1 is a special case (n = 1) of Theorem 2 (Theorem 3, resp.) in [5]. In Section4 we prove the following. Theorem 2. Let f : R ! R be a smooth function. For a rational function j(y) in y, we let g denote the function 2 defined by g(x, y) = f (x) + j(y). Suppose that the level curves Xk(k 2 Sg) of g in the plane E are strictly convex. Then the following conditions are equivalent. 1. The function g satisfies (A∗). 3 2. For k 2 Sg, k(p)jrg(p)j = c(k) is constant on Xk, where k(p) denotes the curvature of Xk at p 2 Xk. 3. Both of the functions j(y) and f (x) are quadratic. Hence, each Xk is a conic section. When the function g is homogeneous, in Section5 we prove the following characterization theorem for conic sections. 2 Theorem 3. Let g : R ! R be a smooth homogeneous function of degree d. Suppose that the level curves Xk 2 of g with k 2 Sg in the plane E are strictly convex. Then the following conditions are equivalent. 1. The function g satisfies (A∗). 3 2. For k 2 Sg, k(p)jrg(p)j = c(k) is constant on Xk, where k(p) denotes the curvature of Xk at p 2 Xk. 3. The function g is given by g(x, y) = (ax2 + 2hxy + by2)d/2, 2 where a, b and h satisfy ab − h 6= 0. Thus, each Xk is either a hyperbola or an ellipse centered at the origin. Finally, we prove the following in Section6. Proposition 2. There exists a function g(x, y) = f (x) + j(y) which satisfies the following. 1. Every level curve of g is strictly convex with Sg = R. 3 2. For k 2 Sg, k(p)jrg(p)j = c(k) is constant on Xk, where k(p) denotes the curvature of Xk at p 2 Xk. 3. The function g does not satisfy (A∗). A lot of properties of conic sections (especially, parabolas) have been proved to be characteristic ones [6–13].
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